## 09. Research topics for Juniors & Seniors

### CAPSTONE RESEARCH PROJECTS

The major expectation for our upper level students is the carrying out of a substantial research project to the point where the results and methods can be presented at a professional meeting (SIAM, MAA, AMS, MathFest) and written up for publication in a refereed research journal. Not all students will reach this level of attainment – yet all will aim for it. Deep and extensive faculty mentoring, and sustained group support, will see most students reach this level in their research.

Students will begin a substantial year-long research project in the second semester of their junior year through the first semester of their senior year. In the second semester of their senior year they will write up and present their work.

Examples of research in progress, recently completed research, and new research topics, include the following:

- High order approximation of the soliton solution of the traffic equation
- Radial basis function neural network model for discontinuous signals
- Numerical analysis of difference equations from loop quantum cosmology
- Fractal properties of numerical convergence for the logistic map
- Numerical approximation of the magnetized proton exchange membrane fuel cell
- Stochastic modeling of the viability analysis of tiger populations
- Epidemiological modeling of avian flu in human population
- Application of genetic algorithms to gene interactive rules to design a process for preventing the formation of cancerous cells
- A spectral collocation method for singularly perturbed hyperbolic equations and its application to a point particle around a Schwarzschild black hole
- Distribution of zeros of derivatives of characteristic polynomials of unitary and orthogonal matrices
- Eigenvalue spacings for regular graphs
- Power laws and heavy-tailed distributions of words in literary texts
- Birth and death processes in graphs and networks
- Multi-type branching processes with dependent offspring
- The parking problem
- Ergodic theory and visualization
- Pattern formation in growing sandpiles